# Seminario Aritmética y Geometría en Valparaíso

## Diciembre

Viernes 5 de Diciembre, 14:30-16:40, sala 2-1, IMA de la PUCV

Expositor: Bas Edixhoven (U. Leiden)

Título: Modular forms, Galois representations, computations, sums of squares and Gauss’s theorem on sums of 3 squares.

Resumen: The first half of the lecture will consist of an overview of joint work with Couveignes, de Jong, Merkl, and of work of Bruin, on the computation of Galois representations attached to modular forms. A nice application of this is the fast computation of the number of solutions of x_1^2+….+x_d^2 = n for even d. The second half of the lecture concerns the case d=3. Gauss has shown for example that for a positive square free integer n that is 1 modulo 4 the number of solutions in integers of x^2+y^2+z^2=n equals 12 times the class number of the ring Z[t]/(t^2+n). Gauss’s proof is long. The aim of the talk is to give a short proof, using the action of the group scheme SO(3). This proof shows in fact that the class group in question acts freely and transitively of the set of solutions. These results on Gauss’s theorem is work in progress of Albert Gunawan, PhD student with me and Qing Liu. Basic knowledge of sheaves on Spec(Z) with its Zariski topology suffices for understanding this second half. Reference for the first half: http://www.math.u-bordeaux1.fr/~jcouveig/book.htm

Anuncios

## Noviembre

Martes  18 de Noviembre,  sala 2-3, IMA de la PUCV, 15:40-16:40

Expositor: Pierre Gillibert (PUCV)

Título:  Kuratowski’s characterisation of the Aleph

Resumen: I shall investigate an infinite combinatorial statement given by Kuratowski to characterize (small) infinite cardinals. After some historical background that give insight, and yeld to this statement, I shall show relations with some undecidable statements and talk about generalisations.

Martes  4 de Noviembre,  sala 2-3, IMA de la PUCV, 15:40-17:00

Expositor: Gabriele Ranieri (PUCV)

Resumen:   Let $k$  be a number field and let $\mathcal{A}$ be a
commutative algebraic group defined over $k$. Consider the following
question:

Problem. Let $P \in \mathcal{A} ( k )$ and let $q$ be a positive
integer. Suppose that for all but finitely many places $v$ of $k$,
there exists $D_v \in \mathcal{A} ( k_v )$ such that $P = q D_v$. Does
there exist $D \in \mathcal{A} ( k )$ such that $P = qD$?

This problem is called Local-global divisibility problem by $q$ on
$\mathcal{A} ( k )$.

Dvornicich and Zannier gave a cohomological interpretation of the
Local-global divisibility problem.
By using this interpretation, in two joint works with Laura Paladino
(University of Calabria) and Evelina Viada (University of Goettingen),
we studied the problem in the case when $\mathcal{A}$ is an elliptic
curve.
Recently, we have partially extended our results on elliptic curves to
the more general family of $GL_2$-type varieties.

We give an idea of the proof of our results and we explain some other
possible generalizations.